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Abstract: Abstract The aim of this article is not to study any practical design for a breakwater device nor to show the evidence of a particular event when waves break over a varying bathymetry, but to promote a paper showing an interesting idea of wave decomposition prior to impact, used in an experimental and numerical study published by Yasuda et al. (Proceedings 25th international conferences coastal engineering, pp 300–313, 1996) and Yasuda et al. (Coast Eng J 41(2): 269–280, 1999. We investigated the new type of breaker, proposed by Yasuda et al. (1996), by detailing several geometric aspects which lead to the unusual size and behavior of some very large plunging jets generated when waves break above some drastic changes of bathymetry. We thoroughly investigated all geometrical aspects of the breaking process, to propose a classification of the breaker types which were observed in our numerical results. We indicated the influences of the reef parameters (steps heights and lengths) on the subsequent breaking process. We also showed that the air entrainment was indeed much larger during the composite breaker occurrence. PubDate: 2022-08-01

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Abstract: Abstract We consider the issue of wave-breaking closure for the well-known Green–Naghdi model and attempt at providing some more understanding of the sensitivity of some closure approaches to the numerical set-up. More precisely and based on Kazolea and Ricchiuto (Ocean Model 123:16–39, 2018), we used two closure strategies for modeling wave-breaking of a solitary wave over a slope. The first one is the hybrid method consisting of suppressing the dispersive terms in a breaking region and the second one is an eddy viscosity approach based on the solution of a turbulent kinetic energy model. The two closures use the same conditions for the triggering of the breaking mechanisms. Both the triggering conditions and the breaking models themselves use case depended/ad/hoc parameters which are affecting the numerical solution while changing. The scope of this work is to make use of sensitivity indices computed by means of analysis of variance to provide the sensitivity of wave-breaking simulation to the variation of parameters such as the mesh size and the breaking parameters specific to each breaking model. The sensitivity analysis is performed using the UQlab framework for uncertainty quantification (Marelli et al., UQLab user manual—sensitivity analysis, Technical report, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Switzerland, 2019). PubDate: 2022-07-26

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Abstract: Abstract Numerical models used in coastal engineering must be computationally efficient while maintaining accuracy at an engineering level. One major weakness of models based on non-hydrostatic Reynolds Averaged Navier–Stokes (RANS) equations is the solution of the Poisson equation used to evolve pressure values over time, accounting for up to 70% of the computational time. We propose a method that uses a differential grid to reduce the computational costs while maintaining a good accuracy: a “subgrid approach”, in which pressure values are computed on a coarse grid while velocities are solved on a finer grid. In addition, the grid resolution is increased near the surface and bottom boundaries. This approach is applied to Iravani et al. (Coast Eng 159:103717, 2020. https://doi.org/10.1016/j.coastaleng.2020.103717) model, which also implements novel free surface boundary conditions for breaking waves. The results show how the computational effort can be reduced up to 70% while providing more than satisfactory results for properties of interest for the coastal engineering practice, like wave height decay and mean water elevation. PubDate: 2022-07-26

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Abstract: Abstract We report on direct numerical simulations of head-on collisions between internal solitary waves in a continuously stratified fluid on the laboratory scale. We find that a complex interplay between the reversal of wave-induced horizontal currents during the collision, and wave-induced vertical currents that act to suction fluid from the boundary layer can lead to significant transport of passive tracers across the boundary layer. We demonstrate that no-slip boundary conditions are essential for this phenomenon to occur, outline the mechanism for the cross-boundary layer transport, and explore how changes in the wave amplitude and stratification affect the efficiency of the process. Finally, we comment on how the results may scale up from the laboratory to the field scale. PubDate: 2022-07-11

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Abstract: Abstract The linear instability study of the viscous Rayleigh–Taylor model in the neighborhood of a laminar smooth increasing density profile \(\rho _0(x_3)\) amounts to the study of the following ordinary differential equation of order 4 0.1 $$\begin{aligned} -\lambda ^2 [ \rho _0 k^2 \phi - (\rho _0 \phi ')'] = \lambda \mu (\phi ^{(4)} - 2k^2 \phi '' + k^4 \phi ) - gk^2 \rho _0'\phi , \end{aligned}$$ where \(\lambda \) is the growth rate in time, and k is the wave number transverse to the density profile. In the case of \(\rho '_0\ge 0\) compactly supported, we provide a spectral analysis showing that in accordance with the results of Helffer and Lafitte (Asymptot Anal 33:189–235, 2003), there is an infinite sequence of non-trivial solutions \((\lambda _n, \phi _n)\) of (0.1), with \(\lambda _n\rightarrow 0\) when \(n\rightarrow +\infty \) and \(\phi _n\in H^4({\mathbf {R}})\) . In the more general case where \(\rho _0'>0\) everywhere and \(\rho _0\) converges at \(\pm \infty \) to finite limits \(\rho _{\pm }>0\) , we prove that there exist finitely non-trivial solutions \((\lambda _n, \phi _n)\) of (0.1). The line of investigation is to reduce both cases to the study of a self-adjoint operator on a compact set. PubDate: 2022-07-04

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Abstract: Abstract In this paper, we introduce a numerical method for approximating the dispersive Serre–Green–Naghdi equations with topography using continuous finite elements. The method is an extension of the hyperbolic relaxation technique introduced in Guermond et al. (J Comput Phys 450:110809, 2022). It is explicit, second-order accurate in space, third-order accurate in time, and is invariant-domain preserving. It is also well balanced and parameter free. Special attention is given to the convex limiting technique when physical source terms are added in the equations. The method is verified with academic benchmarks and validated by comparison with laboratory experimental data. PubDate: 2022-06-27

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Abstract: Abstract In this paper, we study some theoretical and numerical issues of the Boussinesq/Full dispersion system. This is a three-parameter system of pde’s that models the propagation of internal waves along the interface of two-fluid layers with rigid lid condition for the upper layer, and under a Boussinesq regime for the upper layer and a full dispersion regime for the lower layer. We first discretize in space the periodic initial-value problem with a Fourier–Galerkin spectral method and prove error estimates for several ranges of values of the parameters. Solitary-wave solutions of the model are then studied numerically in several ways. The numerical generation is analyzed by approximating the ode system with periodic boundary conditions for the solitary-wave profiles with a Fourier spectral scheme, implemented in a collocation form, and solving iteratively the corresponding algebraic system in Fourier space with the Petviashvili method accelerated with the minimal polynomial extrapolation technique. Motivated by the numerical experiments, a new result of existence of solitary waves is proved. In the last part of the paper, the dynamics of these solitary waves is studied computationally, To this end, the semidiscrete systems obtained from the Fourier–Galerkin discretization in space are integrated numerically in time by a Runge–Kutta Composition method of order four. The fully discrete scheme is used to explore numerically the stability of solitary waves, their collisions, and the resolution of other initial conditions into solitary waves. PubDate: 2022-06-27

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Abstract: Abstract One of the features of Boussinesq-type models for dispersive wave propagation is the presence of mixed spatial/temporal derivatives in the partial differential system. This is a critical point in the design of the time marching strategy, as the cost of inverting the algebraic equations arising from the discretization of these mixed terms may result in a nonnegligible overhead. In this paper, we propose novel approaches based on the classical Lax–Wendroff (LW) strategy to achieve single-step high-order schemes in time. To reduce the cost of evaluating the complex correction terms arising in the Lax–Wendroff procedure for Boussinesq equations, we propose several simplified strategies which allow to reduce the computational time at fixed accuracy. To evaluate these qualities, we perform a spectral analysis to assess the dispersion and damping error. We then evaluate the schemes on several benchmarks involving dispersive propagation over flat and nonflat bathymetries, and perform numerical grid convergence studies on two of them. Our results show a potential for a CPU reduction between 35 and 40% to obtain accuracy levels comparable to those of the classical RK3 method. PubDate: 2022-06-10

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Abstract: Abstract The linear stability of the laminar boundary layer flow of a Stokes wave in deep waters is investigated by means of a ‘momentary’ criterion of instability for unsteady flows. In the parameter range investigated, it is found that the flow is stable to 2-D perturbations. The least stable eigenmode of the resulting Orr–Sommerfeld spectrum attains its maximum beneath the boundary layer of the Stokes wave. The laminar boundary layer flow of a Stokes wave appears to be stable to infinitesimal perturbations at any Reynolds numbers, but it may be unstable to finite perturbations as in Poiseuille pipe flows. The present study is motivated by the recent experimental evidence of spontaneous occurrence of turbulence beneath unforced non-breaking surface waves. PubDate: 2022-06-07

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Abstract: Abstract Extreme water-wave motion is investigated analytically and numerically by considering two-soliton and three-soliton interactions on a horizontal plane. We successfully determine numerically that soliton solutions of the unidirectional Kadomtsev–Petviashvili equation (KPE), with equal far-field individual amplitudes, survive reasonably well in the bidirectional and higher-order Benney–Luke equations (BLE). A well-known exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed the BLE at an initial time, and we confirm that the KPE-fourfold amplification approximately persists. More interestingly, a known three-soliton solution of the KPE is analysed further to assess its eight- or ninefold amplification, the latter of which exists in only a special and difficult-to-attain limit. This solution leads to an extreme splash at one point in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time to establish the maximum amplification: it is approximately 7.8 for a KPE amplification of 8.4. Herein, the computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to cnoidal-wave solutions. Moreover, special geometric (finite-element) variational integrators in space and time have been used in order to eradicate artificial numerical damping of, in particular, wave amplitude. PubDate: 2022-06-02

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Abstract: Abstract A nonlinear evolution equation is derived for surface gravity waves correct up to fourth order in wave steepness in the presence of a pycnocline of finite thickness in a fluid domain of infinite depth. Using this nonlinear evolution equation, the stability analysis is performed for a uniform-wave train solution. It is observed that pycnocline of finite thickness has less stabilizing influence than thin pycnocline. The maximum growth rate of instability becomes less due to presence of pycnocline, and it increases as the thickness of the pycnocline increases. It is revealed that the maximum growth rate of instability decreases as the density differences across the pycnocline increases and also decreases as the depth of the pycnocline increases. The maximum growth rate of instability becomes higher for smaller values of the angle between the direction of the space variation of the amplitudes and the direction of propagation of the wave. Also, it is observed that the unstable region increases with the decrease in the depth of the pycnocline. PubDate: 2022-04-12 DOI: 10.1007/s42286-022-00058-4

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Abstract: Abstract The focus here is upon the generalized Korteweg–de Vries equation, $$\begin{aligned} u_t + u_x + \frac{1}{p} \left( u^p \right) _x +u_{xxx} \, = \, 0, \end{aligned}$$ where \(p = 2, 3, \ldots \) . When \(p \ge 5\) , it is thought that the equation is not globally well posed in time for \(L_2\) -based Sobolev class data. Various numerical simulations carried out by multiple research groups indicate that solutions can blowup in finite time for large, smooth initial data. This is known to be the case in the critical case \(p = 5\) , but remains a conjecture for supercritical values of p. Studied here are methods for controlling this potential blow up. Several candidates are put forward; the addition of dissipation or of higher order dispersion are two obvious candidates. However, these apparently can only work for a limited range of nonlinearities. However, the introduction of high frequency temporal oscillations appear to be more effective. Both temporal oscillation of the nonlinearity and of the boundary condition in an initial-boundary-value configuration are considered. The bulk of the discussion will turn around this prospect in fact. PubDate: 2022-04-11 DOI: 10.1007/s42286-022-00057-5

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Abstract: Abstract Understanding the droplet cloud and spray dynamics is important for the study of the ocean surface and marine boundary layer. The role that the wave energy and the type of wave breaking play in the resulting distribution and dynamics of droplets are yet to be understood. The aim of this work was to generate violent plunging breakers in the laboratory and analyze the spray production post-breaking, i.e. after the crest of the wave impacts in the free surface. The droplet sizes and their dynamics were measured with imaging techniques and the effect of different wind speeds on the droplet production was also considered. It was found that the mean radius increases with the wave energy content and the number of large droplets (radius > 1 mm) in the vertical direction increases with the presence of wind. Furthermore, the normalized distribution of droplet sizes is consistent with the distribution of ligament-mediated spray formation. Also, indications of turbulence affecting the droplet dynamics at wind speeds of 5 m/s were found. The amount of large droplets (radius > 1 mm) found in this work was larger than reported in field studies. PubDate: 2022-03-31 DOI: 10.1007/s42286-022-00056-6

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Abstract: Abstract If a wavemaker at one end of a water-wave tank oscillates with a particular frequency, time series of downstream surface waves typically include that frequency along with its harmonics (integer multiples of the original frequency). This behavior is common for the propagation of weakly nonlinear waves with a narrow band of frequencies centered around the dominant frequency such as in the evolution of ocean swell, pulse propagation in optical fibers, and Bose-Einstein condensates. Presented herein are measurements of the amplitudes of the first and second harmonic bands from four surface water wave laboratory experiments. The Stokes expansion for small-amplitude surface water waves provides predictions for the amplitudes of the second and higher harmonics given the amplitude of the first harmonic. Similarly, the derivations of the NLS equation and its generalizations (models for the evolution of weakly nonlinear, narrow-banded waves) provide predictions for the second and third harmonic bands given amplitudes of the first harmonic band. We test the accuracy of these predictions by making two types of comparisons with experimental measurements. First, we consider the evolution of the second harmonic band while neglecting all other harmonic bands. Second, we use explicit Stokes and generalized NLS formulas to predict the evolution of the second harmonic band using the first harmonic data as input. Comparisons of both types show reasonable agreement, though predictions obtained from dissipative generalizations of NLS consistently outperform the conservative ones. Finally, we show that the predictions obtained from these two methods are qualitatively different. PubDate: 2022-02-15 DOI: 10.1007/s42286-022-00055-7

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Abstract: Abstract We study the spectral stability of small-amplitude periodic traveling waves of the Ostrovsky equation. We prove that these waves exhibit spectral instabilities arising from a collision of pair of non-zero eigenvalues on the imaginary axis when subjected to square-integrable perturbations on the whole real line. We also list all such collisions between pairs of eigenvalues on the imaginary axis and do a Krein signature analysis. PubDate: 2022-01-12 DOI: 10.1007/s42286-021-00054-0

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Abstract: Abstract In this article, we provide formulations of energy flux and radiation stress consistent with the scaling regime of the Korteweg–de Vries (KdV) equation. These quantities can be used to describe the shoaling of cnoidal waves approaching a gently sloping beach. The transformation of these waves along the slope can be described using the shoaling equations, a set of three nonlinear equations in three unknowns: the wave height H, the set-down \({\bar{\eta }}\) and the elliptic parameter m. We define a numerical algorithm for the efficient solution of the shoaling equations, and we verify our shoaling formulation by comparing with experimental data from two sets of experiments as well as shoaling curves obtained in previous works. PubDate: 2021-10-25 DOI: 10.1007/s42286-021-00053-1

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Abstract: Abstract In this contribution, we prove that small amplitude, resonant harmonic, spatially periodic traveling waves (Wilton ripples) exist in a family of weakly nonlinear PDEs which model water waves. The proof is inspired by that of Reeder and Shinbrot (Arch. Rat. Mech. Anal. 77:321–347, 1981) and complements the authors’ recent, independent result proven by a perturbative technique (Akers and Nicholls 2021). The method is based on a Banach Fixed Point Iteration and, in addition to proving that this iteration has Wilton ripples as a fixed point, we use it as a numerical method for simulating these solutions. The output of this numerical scheme and its performance are evaluated against a quasi-Newton iteration. PubDate: 2021-08-05 DOI: 10.1007/s42286-021-00052-2

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Abstract: Abstract One hundred years ago, Nekrasov published the widely cited paper (Nekrasov in Izvestia Ivanovo-Voznesensk Politekhn Inst 3:52–65, 1921), in which he derived the first of his two integral equations describing steady periodic waves on the free surface of water. We examine how Nekrasov arrived at these equations and his approach to investigating their solutions. In connection with this, Nekrasov’s life after 1917 is briefly outlined, in particular, how he became a prisoner in Stalin’s Gulag. Further results concerning Nekrasov’s equations and related topics are surveyed. PubDate: 2021-06-24 DOI: 10.1007/s42286-021-00051-3

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Abstract: Abstract It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent. PubDate: 2021-01-06 DOI: 10.1007/s42286-020-00046-6

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Abstract: Abstract This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the solutions. Furthermore, it is uniform both in the infinite depth limit and the zero surface tension limit. PubDate: 2020-11-02 DOI: 10.1007/s42286-020-00044-8